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A159677
Expansion of 64*x^2/(1 - 1023*x + 1023*x^2 - x^3).
3
0, 0, 64, 65472, 66912384, 68384391040, 69888780730560, 71426265522241344, 72997573474949923072, 74603448665133299138304, 76244651538192756769423680, 77921959268584332285051862720, 79636166127841649402566234276224, 81388083860694897105090406378438272
OFFSET
0,3
COMMENTS
Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 15*n(j) + 1 = a(j)*a(j) and 17*n(j) + 1 = b(j)*b(j) with positive integer numbers.
FORMULA
The a(j) recurrence is a(0)=1, a(1)=31, a(t+2) = 32*a(t+1) - a(t) resulting in terms 1, 31, 991, 31681, ... (A159674).
The b(j) recurrence is b(0)=1, b(1)=33, b(t+2) = 32*b(t+1) - b(t) resulting in terms 1, 33, 1055, 33727, ... (A159675).
The n(j) recurrence is n(-1) = n(0) = 0, n(1) = 64, n(t+3) = 1023*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 64, 65472, 66912384, ... (this sequence).
a(n) = -((511+32*sqrt(255))^(-n)*(-1+(511+32*sqrt(255))^n)*(16+sqrt(255)+(-16+sqrt(255))*(511+32*sqrt(255))^n))/510. - Colin Barker, Jul 25 2016
a(n) = (16/255)*(-1 + ChebyshevU(n, 511) - 1021*ChebyshevU(n-1, 511)). - G. C. Greubel, Sep 25 2022
MAPLE
for a from 1 by 2 to 100000 do b:=sqrt((17*a*a-2)/15): if (trunc(b)=b) then
n:=(a*a-1)/15: La:=[op(La), a]:Lb:=[op(Lb), b]:Ln:=[op(Ln), n]: endif: enddo:
MATHEMATICA
RecurrenceTable[{a[0]==a[1]==0, a[2]==64, a[n]==1023(a[n-1]-a[n-2])+ a[n-3]}, a, {n, 20}] (* Harvey P. Dale, Jan 01 2014 *)
LinearRecurrence[{1023, -1023, 1}, {0, 0, 64}, 20] (* Harvey P. Dale, Jan 01 2014 *)
PROG
(PARI) concat([0, 0], Vec(64/(-x^3+1023*x^2-1023*x+1) + O(x^20))) \\ Colin Barker, Mar 04 2014
(PARI) a(n) = round(-((511+32*sqrt(255))^(-n)*(-1+(511+32*sqrt(255))^n)*(16+sqrt(255)+(-16+sqrt(255))*(511+32*sqrt(255))^n))/510) \\ Colin Barker, Jul 25 2016
(Magma) I:=[0, 0, 64]; [n le 3 select I[n] else 1023*Self(n-1) - 1023*Self(n-2) +Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 03 2018
(SageMath)
def A159677(n): return (16/255)*(-1 +chebyshev_U(n, 511) -1021*chebyshev_U(n-1, 511))
[A159677(n) for n in range(31)] # G. C. Greubel, Sep 25 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Apr 19 2009
EXTENSIONS
More terms from Harvey P. Dale, Jan 01 2014
New name from Colin Barker, Feb 24 2014
Offset changed to 0 by Colin Barker, Mar 04 2014
STATUS
approved